asymptotic expansions as ϵ→0

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11: 2.9 Difference Equations

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$f(n)\sim {\displaystyle \sum _{s=0}^{\mathrm{\infty }}}{\displaystyle \frac{f_s}{n^s}},$

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$g(n)\sim {\displaystyle \sum _{s=0}^{\mathrm{\infty }}}{\displaystyle \frac{g_s}{n^s}}$ , $n\to \mathrm{\infty }$,

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2.9.8 $$w_j(n)\sim {\rho }_j^n{n}^{{\alpha }_j}\sum _{s=0}^{\mathrm{\infty }}\frac{a_{s,j}}{n^s},$$ $n\to \mathrm{\infty }$.

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Symbols:

$\sim$: Poincaré asymptotic expansion, $n$: nonnegative integer, $a_{s,j}$: coefficients, ${\rho }_j$: roots and $w_j(n)$: solutions

Permalink:

http://dlmf.nist.gov/2.9.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §2.9(i), §2.9 and Ch.2

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2.9.12 $$w_j(n)\sim {\rho }^n{n}^{{\alpha }_j}\sum _{s=0}^{\mathrm{\infty }}\frac{a_{s,j}}{n^s},$$ $n\to \mathrm{\infty }$,

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\rho$: roots and $w_j(n)$: solutions

Referenced by:

§2.9(ii)

Permalink:

http://dlmf.nist.gov/2.9.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §2.9(ii), §2.9 and Ch.2

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2.9.13 $$w_2(n)\sim {\rho }^n{n}^{{\alpha }_2}\sum _{\begin{array}{c}s=0\\ s\ne {\alpha }_2-{\alpha }_1\end{array}}^{\mathrm{\infty }}\frac{b_s}{n^s}+c{w}_1(n)\ln n,$$ $n\to \mathrm{\infty }$.

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\ln z$: principal branch of logarithm function, $\rho$: roots, $b_s$: coefficients, $c$: constant and $w_j(n)$: solutions

Permalink:

http://dlmf.nist.gov/2.9.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §2.9(ii), §2.9 and Ch.2

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12: 29.7 Asymptotic Expansions

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29.7.1 $$a_{\nu }^m\left(k^2\right)\sim p\kappa -{\tau }_0-{\tau }_1{\kappa }^{-1}-{\tau }_2{\kappa }^{-2}-\mathrm{\cdots },$$

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Symbols:

$a_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $\sim$: Poincaré asymptotic expansion, $m$: nonnegative integer, $p$: nonnegative integer, $k$: real parameter, $\nu$: real parameter, $\kappa$ and ${\tau }_j$: coefficients

Permalink:

http://dlmf.nist.gov/29.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §29.7(i), §29.7 and Ch.29

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13: 14.15 Uniform Asymptotic Approximations

… ►In other words, the convergent hypergeometric series expansions of ${𝖯}_{\nu }^{-\mu }\left(\pm x\right)$ are also generalized (and uniform) asymptotic expansions as $\mu \to \mathrm{\infty }$, with scale $1/\mathrm{\Gamma }\left(j+1+\mu \right)$, $j=0,1,2,\mathrm{\dots }$; compare §2.1(v). …

14: 2.4 Contour Integrals

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2.4.1 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}q(t)dt\sim \sum _{s=0}^{\mathrm{\infty }}\mathrm{\Gamma }\left(\frac{s+\lambda }{\mu }\right)\frac{a_s}{z^{(s+\lambda )/\mu }}$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\lambda$: constant, $a$: left endpoint, $q(t)$: analytic function and $\mu$: positive constant

Referenced by:

§2.4(i)

Permalink:

http://dlmf.nist.gov/2.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §2.4(i), §2.4 and Ch.2

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2.4.4 $$Q(z)\sim \sum _{s=0}^{\mathrm{\infty }}\mathrm{\Gamma }\left(\frac{s+\lambda }{\mu }\right)\frac{a_s}{z^{(s+\lambda )/\mu }},$$ $z\to \mathrm{\infty }$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\lambda$: constant, $a$: left endpoint, $\mu$: positive constant and $Q(z)$: Laplace transform of $q(t)$

Referenced by:

§2.4(ii)

Permalink:

http://dlmf.nist.gov/2.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §2.4(ii), §2.4 and Ch.2

… ►Furthermore, as $t\to 0+$, $q(t)$ has the expansion (2.3.7). … ►If $p^{\prime }(t_0)\ne 0$, then $\mu =1$, $\lambda$ is a positive integer, and the two resulting asymptotic expansions are identical. … ►

2.4.15 $${\int }_a^b{\mathrm{e}}^{-zp(t)}q(t)dt\sim 2{\mathrm{e}}^{-zp(t_0)}\sum _{s=0}^{\mathrm{\infty }}\mathrm{\Gamma }\left(s+\frac{1}{2}\right)\frac{b_{2s}}{z^{s+(1/2)}},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $a$: left endpoint, $b$: right endpoint, $p(t)$: analytic function and $q(t)$: analytic function

Referenced by:

§2.4(iv)

Permalink:

http://dlmf.nist.gov/2.4.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §2.4(iv), §2.4 and Ch.2

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15: 10.67 Asymptotic Expansions for Large Argument

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10.67.1 $${\ker }_{\nu }x\sim {\mathrm{e}}^{-x/\sqrt{2}}{\left(\frac{\pi }{2x}\right)}^{\frac{1}{2}}\sum _{k=0}^{\mathrm{\infty }}\frac{a_k(\nu )}{x^k}\cos \left(\frac{x}{\sqrt{2}}+\left(\frac{\nu }{2}+\frac{k}{4}+\frac{1}{8}\right)\pi \right),$$

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Symbols:

${\ker }_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter and $a_k(\nu )$: polynomial coefficient

A&S Ref:

9.10.3, 9.10.5--9.10.7 (modified)

Referenced by:

§10.61(v), §10.67(ii), §10.68(iii)

Permalink:

http://dlmf.nist.gov/10.67.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.67(i), §10.67 and Ch.10

►

10.67.2 $${\mathrm{kei}}_{\nu }x\sim -{\mathrm{e}}^{-x/\sqrt{2}}{\left(\frac{\pi }{2x}\right)}^{\frac{1}{2}}\sum _{k=0}^{\mathrm{\infty }}\frac{a_k(\nu )}{x^k}\sin \left(\frac{x}{\sqrt{2}}+\left(\frac{\nu }{2}+\frac{k}{4}+\frac{1}{8}\right)\pi \right).$$

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Symbols:

${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\sin z$: sine function, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter and $a_k(\nu )$: polynomial coefficient

A&S Ref:

9.10.4, 9.10.5--9.10.7 (modified)

Referenced by:

§10.61(v)

Permalink:

http://dlmf.nist.gov/10.67.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.67(i), §10.67 and Ch.10

… ►

10.67.5 $${\ker }_{\nu }^{\prime }x\sim -{\mathrm{e}}^{-x/\sqrt{2}}{\left(\frac{\pi }{2x}\right)}^{\frac{1}{2}}\sum _{k=0}^{\mathrm{\infty }}\frac{b_k(\nu )}{x^k}\cos \left(\frac{x}{\sqrt{2}}+\left(\frac{\nu }{2}+\frac{k}{4}-\frac{1}{8}\right)\pi \right),$$

ⓘ

Symbols:

${\ker }_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter and $b_k(\nu )$: polynomial coefficient

A&S Ref:

§9.10 (modified)

Permalink:

http://dlmf.nist.gov/10.67.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.67(i), §10.67 and Ch.10

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10.67.6 $${\mathrm{kei}}_{\nu }^{\prime }x\sim {\mathrm{e}}^{-x/\sqrt{2}}{\left(\frac{\pi }{2x}\right)}^{\frac{1}{2}}\sum _{k=0}^{\mathrm{\infty }}\frac{b_k(\nu )}{x^k}\sin \left(\frac{x}{\sqrt{2}}+\left(\frac{\nu }{2}+\frac{k}{4}-\frac{1}{8}\right)\pi \right).$$

ⓘ

Symbols:

${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\sin z$: sine function, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter and $b_k(\nu )$: polynomial coefficient

A&S Ref:

§9.10 (modified)

Permalink:

http://dlmf.nist.gov/10.67.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.67(i), §10.67 and Ch.10

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§10.67(ii) Cross-Products and Sums of Squares in the Case $\nu =0$

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16: 33.12 Asymptotic Expansions for Large $\eta$

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33.12.2 $$\genfrac{}{}{0pt}{}{F_0(\eta ,\rho )}{G_0(\eta ,\rho )}\sim {\pi }^{1/2}{(2\eta )}^{1/6}\left\{\genfrac{}{}{0pt}{}{\mathrm{Ai}\left(x\right)}{\mathrm{Bi}\left(x\right)}\left(1+\frac{B_1}{\mu }+\frac{B_2}{{\mu }^2}+\mathrm{\cdots }\right)+\genfrac{}{}{0pt}{}{{\mathrm{Ai}}^{\prime }\left(x\right)}{{\mathrm{Bi}}^{\prime }\left(x\right)}\left(\frac{A_1}{\mu }+\frac{A_2}{{\mu }^2}+\mathrm{\cdots }\right)\right\},$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\rho$: nonnegative real variable, $\eta$: real parameter, $x$: variable, $\mu$: variable, $A_j$: coefficients and $B_j$: coefficients

A&S Ref:

14.4.9

Permalink:

http://dlmf.nist.gov/33.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.12(i), §33.12 and Ch.33

►

33.12.3 $$\genfrac{}{}{0pt}{}{F_0^{\prime }(\eta ,\rho )}{G_0^{\prime }(\eta ,\rho )}\sim -{\pi }^{1/2}{(2\eta )}^{-1/6}\left\{\genfrac{}{}{0pt}{}{\mathrm{Ai}\left(x\right)}{\mathrm{Bi}\left(x\right)}\left(\frac{B_1^{\prime }+x{A}_1}{\mu }+\frac{B_2^{\prime }+x{A}_2}{{\mu }^2}+\mathrm{\cdots }\right)+\genfrac{}{}{0pt}{}{{\mathrm{Ai}}^{\prime }\left(x\right)}{{\mathrm{Bi}}^{\prime }\left(x\right)}\left(\frac{B_1+A_1^{\prime }}{\mu }+\frac{B_2+A_2^{\prime }}{{\mu }^2}+\mathrm{\cdots }\right)\right\},$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\rho$: nonnegative real variable, $\eta$: real parameter, $x$: variable, $\mu$: variable, $A_j$: coefficients and $B_j$: coefficients

A&S Ref:

14.4.10

Permalink:

http://dlmf.nist.gov/33.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.12(i), §33.12 and Ch.33

… ►

33.12.6 $$\genfrac{}{}{0pt}{}{F_0(\eta ,2\eta )}{3^{-1/2}{G}_0(\eta ,2\eta )}\sim \frac{\mathrm{\Gamma }\left(\frac{1}{3}\right){\omega }^{1/2}}{2\sqrt{\pi }}\left(1\mp \frac{2}{35}\frac{\mathrm{\Gamma }\left(\frac{2}{3}\right)}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}\frac{1}{{\omega }^4}-\frac{8}{2025}\frac{1}{{\omega }^6}\mp \frac{5792}{\mathrm{46\hspace{0.17em}06875}}\frac{\mathrm{\Gamma }\left(\frac{2}{3}\right)}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}\frac{1}{{\omega }^{10}}-\mathrm{\cdots }\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\eta$: real parameter and $\omega$: factor

Permalink:

http://dlmf.nist.gov/33.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §33.12(i), §33.12 and Ch.33

►

33.12.7 $$\genfrac{}{}{0pt}{}{F_0^{\prime }(\eta ,2\eta )}{3^{-1/2}{G}_0^{\prime }(\eta ,2\eta )}\sim \frac{\mathrm{\Gamma }\left(\frac{2}{3}\right)}{2\sqrt{\pi }{\omega }^{1/2}}\left(\pm 1+\frac{1}{15}\frac{\mathrm{\Gamma }\left(\frac{1}{3}\right)}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}\frac{1}{{\omega }^2}\pm \frac{2}{14175}\frac{1}{{\omega }^6}+\frac{1436}{\mathrm{23\hspace{0.17em}38875}}\frac{\mathrm{\Gamma }\left(\frac{1}{3}\right)}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}\frac{1}{{\omega }^8}\pm \mathrm{\cdots }\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\eta$: real parameter and $\omega$: factor

Referenced by:

§33.12(i)

Permalink:

http://dlmf.nist.gov/33.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §33.12(i), §33.12 and Ch.33

…

17: 5.11 Asymptotic Expansions

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5.11.3 $$\mathrm{\Gamma }\left(z\right)={\mathrm{e}}^{-z}{z}^z{\left(\frac{2\pi }{z}\right)}^{1/2}{\mathrm{\Gamma }}^{\ast }\left(z\right)\sim {\mathrm{e}}^{-z}{z}^z{\left(\frac{2\pi }{z}\right)}^{1/2}\sum _{k=0}^{\mathrm{\infty }}\frac{g_k}{z^k},$$

ⓘ

Defines:

${\mathrm{\Gamma }}^{\ast }\left(z\right)$: scaled gamma function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $k$: nonnegative integer, $z$: complex variable and $g_k$: coefficients

A&S Ref:

6.1.37

Referenced by:

§10.19(i), §13.8(iv), §5.11(i), §5.11(i), §5.11(i), §5.11(ii), §5.11(ii), §5.21, §5.9(i), §8.11(ii), §8.12, (8.18.13), (8.18.13), §8.18(ii), §8.18(ii), Erratum (V1.1.7) for Additions, Erratum (V1.1.8) for Rearrangement

Permalink:

http://dlmf.nist.gov/5.11.E3

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

The definition of the scaled gamma function ${\mathrm{\Gamma }}^{\ast }\left(z\right)$ was inserted after the first equals sign.

Changes (effective with 1.0.11):

An unnecessary bracket around the sum was removed.

See also:

Annotations for §5.11(i), §5.11 and Ch.5

… ►

5.11.8 $$\mathrm{Ln}\mathrm{\Gamma }\left(z+h\right)\sim \left(z+h-\frac{1}{2}\right)\ln z-z+\frac{1}{2}\ln \left(2\pi \right)+\sum _{k=2}^{\mathrm{\infty }}\frac{{(-1)}^k{B}_k\left(h\right)}{k(k-1)z^{k-1}},$$

ⓘ

Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $z$: complex variable

Referenced by:

§12.10(ii), (5.11.8), §5.11(i), §5.11(ii), Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8), Erratum (V1.0.11) for Equation (5.11.8)

Permalink:

http://dlmf.nist.gov/5.11.E8

Encodings:

pMML, png, TeX

Generalization (effective with 1.0.11):

Originally $h$ in (5.11.8) was unnecessarily restricted to lie in the interval $[0,1]$. In fact, $h$ may lie anywhere in the complex plane.

Suggested 2015-02-28 by Nico Temme

Addition (effective with 1.0.10):

To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to $\mathrm{Ln}\mathrm{\Gamma }\left(z+h\right)$, where $\mathrm{Ln}$ is the general logarithm. Originally $\ln \mathrm{\Gamma }\left(z+h\right)$ was used, where $\ln$ is the principal branch of the logarithm.

Suggested 2015-02-13 by Philippe Spindel

See also:

Annotations for §5.11(i), §5.11(i), §5.11 and Ch.5

… ►

5.11.13 $$\frac{\mathrm{\Gamma }\left(z+a\right)}{\mathrm{\Gamma }\left(z+b\right)}\sim z^{a-b}\sum _{k=0}^{\mathrm{\infty }}\frac{G_k(a,b)}{z^k},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $k$: nonnegative integer, $z$: complex variable, $a$: real or complex variable, $b$: real or complex variable and $G_k(a,b)$: coefficients

A&S Ref:

6.1.47

Permalink:

http://dlmf.nist.gov/5.11.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §5.11(iii), §5.11 and Ch.5

►

5.11.14 $$\frac{\mathrm{\Gamma }\left(z+a\right)}{\mathrm{\Gamma }\left(z+b\right)}\sim {\left(z+\frac{a+b-1}{2}\right)}^{a-b}\sum _{k=0}^{\mathrm{\infty }}\frac{H_k(a,b)}{{\left(z+\frac{1}{2}(a+b-1)\right)}^{2k}}.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $z$: complex variable, $a$: real or complex variable, $b$: real or complex variable and $H_k(a,b)$: coefficients

Referenced by:

§5.11(iii), Erratum (V1.0.21) for Equation (5.11.14)

Permalink:

http://dlmf.nist.gov/5.11.E14

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.21):

The previous constraint $\mathrm{\Re }\left(b-a\right)>0$ was removed, see Fields (1966, (3)).

See also:

Annotations for §5.11(iii), §5.11 and Ch.5

… ►

5.11.19 $$\frac{\mathrm{\Gamma }\left(z+a\right)\mathrm{\Gamma }\left(z+b\right)}{\mathrm{\Gamma }\left(z+c\right)}\sim \sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{{\left(c-a\right)}_k{\left(c-b\right)}_k}{k!}\mathrm{\Gamma }\left(a+b-c+z-k\right).$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $k$: nonnegative integer, $z$: complex variable, $a$: real or complex variable and $b$: real or complex variable

Referenced by:

§5.11(i), §5.11(iii)

Permalink:

http://dlmf.nist.gov/5.11.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §5.11(iii), §5.11 and Ch.5

…

18: 8.20 Asymptotic Expansions of $E_p\left(z\right)$

… ►

8.20.2 $$E_p\left(z\right)\sim \frac{{\mathrm{e}}^{-z}}{z}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{{\left(p\right)}_k}{z^k},$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$,

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\mathrm{ph}$: phase, $z$: complex variable, $p$: parameter, $k$: nonnegative integer and $\delta$: small positive constant

Referenced by:

§8.20(i)

Permalink:

http://dlmf.nist.gov/8.20.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §8.20(i), §8.20 and Ch.8

… ►

8.20.3 $$E_p\left(z\right)\sim \pm \frac{2\pi \mathrm{i}}{\mathrm{\Gamma }\left(p\right)}{\mathrm{e}}^{\mp p\pi \mathrm{i}}{z}^{p-1}+\frac{{\mathrm{e}}^{-z}}{z}\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{\left(p\right)}_k}{z^k},$$ $\frac{1}{2}\pi +\delta \le \pm \mathrm{ph}z\le \frac{7}{2}\pi -\delta$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $z$: complex variable, $p$: parameter, $k$: nonnegative integer and $\delta$: small positive constant

Referenced by:

§2.11(ii), §8.20(i)

Permalink:

http://dlmf.nist.gov/8.20.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.20(i), §8.20 and Ch.8

… ►

8.20.6 $$E_p\left(\lambda p\right)\sim \frac{{\mathrm{e}}^{-\lambda p}}{(\lambda +1)p}\sum _{k=0}^{\mathrm{\infty }}\frac{A_k(\lambda )}{{(\lambda +1)}^{2k}}\frac{1}{p^k},$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $p$: parameter, $k$: nonnegative integer and $A_k(\lambda )$

Permalink:

http://dlmf.nist.gov/8.20.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §8.20(ii), §8.20 and Ch.8

…

19: 8.11 Asymptotic Approximations and Expansions

… ►

8.11.6 $$\gamma (a,z)\sim -z^a{\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{{(-a)}^k{b}_k(\lambda )}{{(z-a)}^{2k+1}},$$ , $|\mathrm{ph}a|\le \frac{\pi }{2}-\delta$.

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $\mathrm{ph}$: phase, $x$: real variable, $z$: complex variable, $a$: parameter, $k$: nonnegative integer, $\delta$: small positive constant, $\lambda$: parameter and $b_k(\lambda )$: coefficients

Notes:

See Paris (2002b)

Referenced by:

§8.11(iii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v)

Permalink:

http://dlmf.nist.gov/8.11.E6

Encodings:

pMML, png, TeX

Addition (effective with 1.0.14):

Originally this equation was stated for real variables $a$ and $x(=\lambda a)$, . It has been extended to allow for complex variables $a$ and $z(=\lambda a)$, where and $|\mathrm{ph}a|\le \frac{\pi }{2}-\delta$.

See also:

Annotations for §8.11(iii), §8.11 and Ch.8

►

8.11.7 $$\mathrm{\Gamma }(a,z)\sim z^a{\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{{(-a)}^k{b}_k(\lambda )}{{(z-a)}^{2k+1}},$$ $\lambda >1$, $|\mathrm{ph}a|\le \frac{3\pi }{2}-\delta$.

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\mathrm{ph}$: phase, $x$: real variable, $z$: complex variable, $a$: parameter, $k$: nonnegative integer, $\delta$: small positive constant, $\lambda$: parameter and $b_k(\lambda )$: coefficients

Referenced by:

§8.11(iii), §8.11(iii), §8.11(iii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v)

Permalink:

http://dlmf.nist.gov/8.11.E7

Encodings:

pMML, png, TeX

Addition (effective with 1.0.14):

Originally this equation was stated for real variables $a$ and $x(=\lambda a)$, $\lambda >1$. It has been extended to allow for complex variables $a$ and $z(=\lambda a)$, where $\lambda >1$ and $|\mathrm{ph}a|\le \frac{3\pi }{2}-\delta$.

See also:

Annotations for §8.11(iii), §8.11 and Ch.8

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8.11.18 $$S_n(x)\sim \sum _{k=0}^{\mathrm{\infty }}{d}_k(x)n^{-k},$$ $n\to \mathrm{\infty }$,

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $x$: real variable, $k$: nonnegative integer, $n$: nonnegative integer, $S_n(x)$: function and $d_k(x)$: coefficients

Referenced by:

§8.11(v)

Permalink:

http://dlmf.nist.gov/8.11.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §8.11(v), §8.11 and Ch.8

…

20: 2.11 Remainder Terms; Stokes Phenomenon

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2.11.6 $$E_p\left(z\right)\sim \frac{{\mathrm{e}}^{-z}}{z}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(p\right)}_s}{z^s}$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm and $E_p\left(z\right)$: generalized exponential integral

Referenced by:

§2.11(ii), §2.11(iii), §2.11(iii), §2.11(iv), §2.11(iv)

Permalink:

http://dlmf.nist.gov/2.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(ii), §2.11 and Ch.2

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2.11.7 $$E_p\left(z\right)\sim \frac{2\pi \mathrm{i}{\mathrm{e}}^{-p\pi \mathrm{i}}}{\mathrm{\Gamma }\left(p\right)}{z}^{p-1}+\frac{{\mathrm{e}}^{-z}}{z}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(p\right)}_s}{z^s},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral and $\mathrm{i}$: imaginary unit

Referenced by:

§2.11(ii), §2.11(iv), §2.11(iv), §2.11(iv)

Permalink:

http://dlmf.nist.gov/2.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(ii), §2.11 and Ch.2

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2.11.13 $$F_{n+p}\left(z\right)\sim \frac{{\mathrm{e}}^{-\mathrm{i}(\rho +\alpha )\theta }}{1+{\mathrm{e}}^{-\mathrm{i}\theta }}\frac{{\mathrm{e}}^{-\rho -z}}{{(2\pi \rho )}^{1/2}}\sum _{s=0}^{\mathrm{\infty }}\frac{a_{2s}(\theta ,\alpha )}{{\rho }^s},$$ $\rho \to \mathrm{\infty }$,

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $F_p\left(z\right)$: terminant function, $\rho$: radius, $\theta$: radius and $a_{2s}(\theta ,\alpha )$: coefficients

Referenced by:

§2.11(iii), §2.11(iii)

Permalink:

http://dlmf.nist.gov/2.11.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(iii), §2.11 and Ch.2

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2.11.24 $${\mathrm{e}}^x{E}_1\left(x\right)\sim \sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{s!}{x^{s+1}},$$ $x\to +\mathrm{\infty }$.

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral and $!$: factorial (as in $n!$)

Referenced by:

§2.11(vi)

Permalink:

http://dlmf.nist.gov/2.11.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(vi), §2.11 and Ch.2

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2.11.29 $$W_{\kappa ,\mu }\left(z\right)\sim \sum _{n=0}^{\mathrm{\infty }}{a}_n,$$

ⓘ

Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion and $a_n$: coefficients

Permalink:

http://dlmf.nist.gov/2.11.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(vi), §2.11 and Ch.2

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